I was recently looking for an activity to explore a linear relationship, preferable one that included some practice with decimals. I found a few examples but none of them really seemed to satisfy what I was looking for. Here are a few of my criteria for good experiments to explore function relationships:

Uses easy to find and inexpensive materials or materials that are commonly found in a mathematics classroom.

Hands on activity for students to investigate in small groups

The instructions are easy to explain and all students have an entry point into the activity.

Since I didn't find quite what I was looking for, I created my own inspired by some of the other examples I found. I ended up doing an activity using a few boxes of paper clips I found in the supply cabinet. I started by asking students to do a notice and wonder about what math questions we could ask using a box of 100 paper clips.

Estimation

I asked students to estimate how long they thought it would take to create a chain of 100 paper clips. I also asked them to think about an estimate that they know was too low (that creating a chain this fast was not possible) and too high (that they would have no problem creating a chain in this time even going slowly). Most students thought that a time between 5 and 6 minutes was a good "just right" estimate.

Gathering Data

Next we gathered some data to test our estimates. A practice round or two is a good idea as students' speed will increase as they figure out a good strategy for chaining the clips together. You might also ask students to do a few trials at each length of chain and take a mean (or perhaps a trimmed mean) to get more accurate data... or you could save this discussion until after students collect some data and then ask them if they feel their data is accurate.

Revising Estimates

After collecting and analyzing some data, I ask students if they'd like to revise their estimate for 100 paper clips. Then we test their revised estimate using a plot of the values they collected and extrapolating. Below is one student's data plotted in Desmos. They estimated 300 seconds (5 minutes) to chain all 100 paperclips.

Nova Scotia Mathematics Curriculum OutcomesMathematics 6 SP01 - Students will be expected to create, label, and interpret line graphs to draw conclusions. Mathematics 6 SP02 - Students will be expected to select, justify, and use appropriate methods of collecting data, including questionnaires, experiments, databases, and electronic media.Mathematics 6 SP03 - Students will be expected to graph collected data and analyze the graph to solve problems. Mathematics 7 PR02 - Students will be expected to create a table of values from a linear relation, graph the table of values, and analyze the graph to draw conclusions and solve problems.Mathematics 7 N02 - Students will be expected to demonstrate an understanding of the addition, subtraction, multiplication and division of decimals to solve problems (for more than one-digit divisors or more than two-digit multipliers, the use of technology is expected). Mathematics 7 SP01 - Students will be expected to demonstrate an understanding of central tendency and range by: determining the measures of central tendency (mean, median, mode) and range; determining the most appropriate measures of central tendency to report findings. Mathematics 7 SP02 - Students will be expected to determine the effect on the mean, median, and mode when an outlier is included in a data set.

This blog post could easily have been titled, "A Long Wait in a Really Long Line" but I like to be positive. I focus on the silver lining instead of the grey cloud. That's why I took a wait in a long long as an opportunity to practice estimation instead of a tedious and boring waste of time.

At the back of the line - 12:06 pm

I spent a sunny Canada Day afternoon at the Halifax Commons. I went with my family to watch the SkyHawks, the Canadian Armed Forces Parachute Team. After the SkyHawks finished their jumps, we headed to the end of a a really long line so that my son could take a turn on a giant inflatable slide. Instead of dwelling on the length of the wait, we decided to focus on some fun estimation questions. How many people do you think are in this line? My son says he thinks it's more than 100 and I agree. How long will it take to get to the front of the line? I feel like we're in for a long wait. My initial estimate is at least 30 minutes. (Note: in order to answer this last question, I took a look at my watch to check the time... 12:06 pm).

Slow progress - 12:27 pm

Have we gotten any closer? It's 12:27 now (21 minutes in line) and it still feels like we've got a long way to go. There appears to be a strong correlation between the age of a child and the likelihood that they will have second thoughts at the top of the slide. This slows the line down dramatically as parents try to coax and cajole their child to make the leap. How many people in this line are kids waiting to slide and how many are parents/guardians? What is the average age of the kids going down the slide?

We're getting close now but this line is SO slow - 12:40 pm

It's 12:40 now and we've been in line for over a half an hour. How many steps do you estimate their are to the top of the slide? I estimate that we're about half the distance to the slide from where we started. I realize that my initial estimate for how long we will be waiting was way off. At this point, I notice a group of upper elementary age students in front of us playing a hand clapping game called Concentration. After a bit, they shift to playing Chopsticks. This is a game I really like and I've used to introduce students to modular arithmetic so I watched their game to check out their strategy. It kept me entertained for a bit.

We finally make it! It is 1:17 pm and we stood in line for 71 minutes. My son seems to think that this was a reasonable investment for such an awesome slide but I have my doubts. At least we got to do a lot of estimation while we waited in line. We definitely won't be heading to the back of the line for a second slide.

I prepared a lesson plan to work with a student. I carefully considered how I would introduce the topic, the path that the lesson might take and the questions that I would ask to prompt our discussion. I thought about the manipulatives that we might use to visualize and physically interact with the problem. I had a course carefully laid out.

I started by drawing an irregular, kidney shaped area on the desk and asked the student how he would estimate the area of the shape. I was prepared for a number of different responses that I thought I might hear... but the student didn't follow my carefully plotted course for our lesson. Instead he replied, "I'd use Pick's Theorem."

I grew up sailing on the Columbia River. When changing course on a sailboat, you can either turn the bow (the front of the boat) through the wind (i.e. tacking) or you can turn the stern (the back of the boat) through the wind (i.e. jibing). When tacking, the boom gently moves from from one side of the boat to the other. Jibing on the other hand can be dangerous as the boom suddenly jumps to the other side of the boat. When the student suggested Pick's Theorem, it felt like changing course by jibing instead of by tacking.

So I altered my plan a bit and we explored Pick's Theorem. I laid out a lattice of dots on top of the shape and the student drew a polygon approximating the shape using the lattice points. We then counted the lattice points on the boundary and the points in the interior to calculate the area of the shape. We came up with an area of 42.5 square units (each square was 3 cm by 3 cm).

After our excursion through Pick's Theorem we found our way back to estimating the area with some manipulatives. First we covered the shape with square tiles and then we covered the shape with pennies.

We found that we could cover the shape with 66 square tiles. I asked the student how the area we found with Pick's Theorem and the area we found with square tiles compared. Through our discussion we decided that we needed a common way to talk about these areas so we converted both to square centimeters. We found that the area from Pick's Theorem was 382.5 cm^2 and the area using square tiles was 412.5 cm^2. Next, we looked at our penny solution. We looked up the diameter of a penny online and found that 135 pennies at 2.85 cm^2 each gave us a total area of 384.75 cm^2. While discussing how this estimate compared to our others, the student started talking about Alex Thue and his theorem on circle packing (this student has a really good memory). The student remembered that the efficiency of hexagonal packed pennies was about 91%. So we used this efficiency to correct our penny estimate to make it even better. This led to another discussion that I hadn't planned on about tesselations and polygons that tile the plane. The student said he had read in a book that there were 14 irregular pentagons that tile the plane. His book was a few years old however so he didn't know that a 15th pentagon had been discovered in 2015 or other recent work in this area.

While the lesson didn't go quite as I had planned, I was really happy to be able to take the student's contributions to the discussion and weave them into the overall narrative of our work. Being flexible, listening to students and incorporating their contributions into a discussion can sometimes throw you off course and you might end up someplace unexpected. The journey along these altered courses however can be incredible.

Over the Christmas holiday, the number of LEGO bricks in my house increased significantly. My son received LEGO sets as gifts from numerous grandparents, aunts and uncles. I was a LEGO fan when I was a child and now I have an excuse to play with them again as an adult. We've had lots of fun recently building sets and designing our own creations. At some point I became inspired to create a scale model of our home.

Planning and Building

I started this small project by building a test model to try out the proportions and to see what kinds of bricks I would need. The sizes of the door and window established the overall size. I continued revising the structure it until it looked right and then started collecting the bricks I needed.

Building this model reminded me of working on an OpenMiddle.com math problem. In an "open middle" problem, there is a one starting point and one solution but many different paths to get to the solution. With LEGO, there are many different ways to create, revise and improve your model. There are lots of different building techniques that will all result in a well designed scale model.

Scale

After I created my initial rough model I did some reading up on LEGO scale. It turns out that it is a fairly complex topic that lots of different people have investigated. I found the Brick Architect web site to be very helpful. For "classic minifigure" scale a ratio of 1:42 can be used. One major difficulty in discussing scale is that the proportions of a LEGO minifigure are not even close to the proportions of an actual person. A LEGO minifigure is about 4 cm tall and 1.6 cm wide. An average male human is about 175 cm tall and 40 cm wide... about half as wide as a minifigure would be at that height. Another challenge is converting units. The architectural drawings of my house are in feet, which I converted to metric (cm), then a scale factor is applied and finally the metric units are converted into LEGO bricks. I found an awesome tool that does this all for you, the LEGO Unit Converter.

Finance

I used a lot of estimation to determine how many bricks of each type I would need. LEGO bricks are not cheap so you don't want to order more than you need (Check out Jon Orr's activity involving cost, Is LEGO Gender Biased?). I purchased the bricks I needed on BrickLink.com, a large online LEGO marketplace. BrickLink provides a detailed price guide for every brick available which makes it really easy to know if you're getting a good deal or not.

I needed lots of 45 degree angle slope bricks for the roof of my house. These price stats let me know what a reasonable price is to pay for new or used bricks of this type. It is amazing to see how many bricks are sold on this site. I think that the stats from this site could make for an interesting grade 12 math research project.

The Finished Project

I only built the front 1/3 of the house for reasons of both space and cost. When I have time, I'll build this model virtually in the LEGO Digital Designer. This tool allows you to design LEGO models in a virtual 3D environment and quickly see the number of bricks used to create it. My son is enjoying our LEGO house and currently has his LEGO fire truck rescuing our LEGO cat from the roof. New adventures await!

​Last summer, my son and I took a bag of coins from our piggy bank to our local TD Bank to cash them in. My son is 5 years old and likes to feed the coins into the machine. No less exciting is taking the proceeds down the street to Woozles, our favourite bookstore. Unfortunately, when we got the the bank, the coin counting machine was gone. The teller told us we would have to roll our own coins from now on.

When I got home, I did a search to find out why the machines were gone and ran across a story from Global News. It turns out that TD Bank had decided to retire all the coin counting machines in Canada in the wake of reports from the U.S. that the machines were short-changing customers. In a segment on the Today Show called 'Rossen Reports', a team investigated the accuracy of a number of Coinstar machines as well as coin-counting machines at various branches of TD Bank. The team tested the accuracy of the machines by carefully preparing bags filled with exactly $300 worth of pennies, nickles, dimes and quarters. They then used the machines to see how close their count was to $300. The Coinstar machines all checked out with the correct $300 total. The TD Bank machines did not fare so well. The totals on the machines tested at 5 different branches were: $299.95, $299.47, $299.30, $296.27 and $256.90. None of the machines returned an accurate $300 count.

I don't think that machines can really be "100% accurate" all the time. What level of accuracy do you think is acceptable from a coin-counting machine? How much time does it take to roll $300 worth of coins and how much is your time worth? I would probably accept $299.95 for the convenience of not having to roll that many coins. I would be a bit more hesitant to accept $296.27 and definitely would not accept $256.90. While the TD Bank machines were free for customers, in Canada, Coinstar machines apply a coin counting fee of 11.9 cents per dollar. For the $300 counted in this test, the fee would have been $35.70. That is a pretty hefty fee.

Questions and Estimations

According to a class action lawsuit filed in New York in April 2016, TD’s coin-counting machines processed 29 billion coins in 2012. Based on this figure and the data collected by the Rossen Report, how much money do you think customers lost? What factors did you consider when making this estimate?

How would you design an experiment to test the accuracy of TD's coin counting machines? Would you test lots of different machines or a few machines multiple times? How many trials would you run to be confident in your results? What factors might contribute to the errors discovered in these machines?

Nova Scotia Mathematics Curriculum OutcomesMathematics 11 S02 - Interpret statistical data, using: confidence intervals, confidence levels and margin of error.Mathematics 11 S03 - Critically analyze society’s use of statistics.Grade 9 SP03 - Students will be expected to develop and implement a project plan for the collection, display, and analysis of data by: formulating a question for investigation; choosing a data collection method that includes social considerations; selecting a population or a sample; collecting the data; displaying the collected data in an appropriate manner; drawing conclusions to answer the question.Grade 7 SP06 - Students will be expected to conduct a probability experiment to compare the theoretical probability (determined using a tree diagram, table, or other graphic organizer) and experimental probability of two independent events.

There are some really big doors around Halifax. The door on Irving Shipbuilding's Halifax Shipyard Assembly and Ultra Hall facility is big enough for large "mega-blocks" of ships under construction to pass through. The doors at IMP Aerospace's Hangar #9 at the Stanfield International airport is big enough for large aircraft to pass through. Which door do you think is the largest?

What Do You Mean by Largest?

The first thing you might want to do is settle on what you mean by "largest". Do you mean width, height, area, mass or some other measurement? Each of these doors might be the largest for a specific measurement. For example, the aircraft hangar door is made of metal and quite probably has more mass than the shipyard door which is constructed of a polyester fabric.

Estimating Dimensions

The shipyard door is really tall but the aircraft hangar door is really wide. Below are pictures of the two facilities from Google earth with the same scale so that you can compare the buildings that these doors are on.

The Discovery Centre is a hands-on science centre in downtown Halifax. My almost 5 year old son and I has visited on numerous occassions. On my most recent visit, my son spent some time building with Lego bricks in the Lindsay Construction Building Centre. While he played, I took a closer look at the Lego replica Town Clock that sits nearby. What would students notice or wonder about given a picture of this model? There is a fantastic sign on the wall behind the model that lists exactly how many Lego bricks of each colour were used in its construction.

Here are a few questions that I thought about:

Does the amount of visible colour help you determine the number of bricks of each colour used to build this model? Since the Lego bricks used are of various sizes, I think estimating the number of bricks (especially white bricks) from the image would be challenging. Perhaps comparing this building to a Lego kit with a know number of pieces (such as the Lego Town Hall) might help.

If I told you that there were about twice as many green bricks as grey bricks, would that help you estimate the number of bricks?

If I told you that this model was build with 12,465 Lego bricks, how many of those bricks would you estimate are white bricks?

Given the number of bricks used, how long do you think it took to build this model?

What is the scale of the model to the actual Town Clock? How precise do you think this scale model is?

Given the list of lego bricks, you could ask students a number of additional questions:

For each different colour of Lego brick, what fraction (or percentage) of the whole model is it? What is the exact fraction and what is the best simple approximation (using a single digit in the numerator).

Can you find two different sets of colours that have similar ratios? Which two sets of colours have the closets ratios?

If you reduced the scale of this model in half, about how many of each color brick would you expect to use?

The Discovery Centre is currently working on a project to build Canada's largest Lego mosaic wall. The wall will be installed at the Discovery Centre's new location when it moves.

Nova Scotia Mathematics Curriculum Outcomes Mathematics 7 - N07 Students will be expected to compare, order, and position positive fractions, positive decimals (to thousandths), and whole numbers by using benchmarks, place value, and equivalent fractions and/or decimals. Mathematics 8 - N03 Students will be expected to demonstrate an understanding of and solve problems involving percents greater than or equal to 0%.Mathematics 8 - N04 Students will be expected to demonstrate an understanding of ratio and rate.Mathematics 9 - N03 Students will be expected to demonstrate an understanding of rational numbers by comparing and ordering rational numbers and solving problems that involve arithmetic operations on rational numbers.Mathematics at Work 11 - G02 Students will be expected to solve problems that involve scale.

I recently read an article on Wired about the Solar Voyager. A pair of engineers, Isaac Penny and Christopher Sam Soon, designed and built an autonomous, solar powered vessel. On June 1st, 2016 the 18 foot vessel, named Solar Voyager set off on its trans-Atlantic adventure from Gloucester, Massachusetts to Portugal, a journey of more than 4800 kilometres. They are predicting that this trip will take 4 months, assuming that there are no catastrophic events mid-Atlantic. One cool thing about this trip is that the Solar Voyager reports it position and other data online every 15 minutes at ​http://www.solar-voyager.com/trackatlantic.html. Currently, about two-weeks into its journey, Solar Voyager is just South of Halifax, Nova Scotia where I live.

Photo from Isaac Penny/The Solar Voyager Team

The image below shows how far the Solar Voyager has traveled during its first two weeks. That is 1/8 of the time estimated for the crossing. Based on the information below, do you think that it will reach its destination in 4 months? What factors did you consider when making your estimation?

Some factors you might consider are currents, weather, equipment malfunction, obstacles/collisions, wear and tear, etc. There are so many variables at play that it must be very hard to make an accurate estimation.

Some Questions/Estimates for Students:

Estimate the probability that Solar Voyager will reach Portugal in 4 months.

Estimate the probability that Solar Voyager completes its journey (i.e. it doesn't sink or malfunction). You could make these predictions and check up on them when you come back to school in the fall.

Do you think that the vessels progress is/will be modeled by a linear function? Is it likely to get faster, slower or progress at the same rate over time?

What is the scale factor of this 18 foot vessel to a typical cargo vessel?

One of the coolest things about this project is that these young engineers "built Solar Voyager in their free time, undertaking this voyage simply for the challenge." How can I commandeer this type of intrinsic motivation for students in math class? What about this project made them want to work so hard "just for the challenge" and not for some extrinsic reward. Was it because they were the ones who selected and designed the task? Did they have just the right skills so that they felt confident that they would be successful? What is something that was relevant to their lives? How did this project captivate their curiosity?

Students and Staff at J.L. Ilsley High School recently returned from a March break trip to Italy. Their stories about Rome and pizza and gelato inspired this "Would You Rather?" math question. Most students are pretty familiar with pizza and have strong opinions to share on their favourite type and restaurant for pizza.

Would you rather have a slice of pizza from New York or from Rome? The New York pizza costs $2.75 US per slice. The Rome pizza costs 1,50€ per 100 grams.

In Rome, pizza by the slice or "pizza al taglio" is typically sold in rectangular pieces by weight. Prices are often listed per 100 grams. Prices can vary greatly depending upon the type and location of restaurant. Restaurants close to major tourist attractions in Rome are often much more expensive. The price I quote above is from Pizza Florida in Rome. Estimating the weight of a typical slice of pizza might be difficult for students. How much does a typical piece of pizza weigh? According to Pizza Pizza, a 1/10 slice of a 14 inch diameter pizza is approximately 110 grams. There is also the issue of currency conversion. You could ask for 3 Euros worth of pizza, but how much will that cost you in Canadian dollars? An online currency conversion website or app can help with currency exchange.

The Nova Scotia Mathematics 10 curriculum has outcomes on both currency exchange and SI to imperial unit measurement conversions so I thought this would be a nice warm up question to be used in that course.

In case you were wondering where you should go to eat pizza, here are the 14 top cities for pizza, as identified on the Conde Nast Traveler Best Pizza in the World list. Note that a Canadian city, Edmonton, made the list.

Chicago, Illinois

New York City, New York

Rome, Italy

Orlando, Florida

Naples, Italy

New Haven, Connecticut

Venice, Italy

Edmonton, Canada

Florence, Italy

Palermo, Italy

Milan, Italy

Philadelphia, Pennsylvania

Los Angeles, California

Las Vegas, Nevada

Nova Scotia Mathematics Curriculum Outcomes Mathematics 10 - M02 Students will be expected to apply proportional reasoning to problems that involve conversions between SI and imperial units of measure. Mathematics 10 - FM01 Students will be expected to solve problems that involve unit pricing and currency exchange, using proportional reasoning.

Revisiting the Classic Ferris Wheel Problem

Just about every textbook with a chapter on graphing sinusoidal functions has an obligatory question about a Ferris wheel. Many of these problems are not particularly engaging and many of them give you all the required information right at the start.

Instead of doing a textbook problem with a fictional Ferris wheel, I decided to use a real Ferris wheel from a nearby amusement park that some of my students would be familiar with. I visited the park to take a video of the Ferris wheel in action. Below is a 30 second clip of the "Big Ellie" Ferris Wheel at Atlantic Playland.

Notice and Wonder

I started by asking students what they noticed in the video. After brainstorming and recording the students observations I asked students what they wondered about in the video. They asked questions like "how fast is the ride going?", "how tall is this Ferris wheel?", "how far can you see from the top of the ride?", "how long does the ride last?". In order to investigate these questions further we needed to estimate some values such as the radius of the wheel, how long it takes to make one revolution, and the height of the central axis about the ground. I asked students to estimate these values using the clues in the video we watched. We watched it several times in order to get some good estimates.

I also talked about some of the mental math required to operate a ride like this. Because it is belt driven, you have to load the Ferris wheel so that it is equally balanced around the wheel. Otherwise, one side of the wheel would become too heavy and the drive cable would slip in the rim and the wheel wouldn't be able to turn! This requires a lot of on the fly estimates of weights of the riders as it is being loaded.

In order to get a see how good we did with our estimations we turned to the internet in order to try to hunt down some of these values with a Google search. This lead to a discussion about what keywords we could use to hunt down this information. A search of "height of the central axis of the Ferris wheel at Atlantic Playland" was not very fruitful... an essential skill to solve a problem like this is to translate mathematical language into common terms that you can use for a Google search. Ve Anusic has a great blog post where he discusses a similar problem and the discussion with his students about the information you need and the information you might find online. First we did a search to find Atlantic Playland's website and found that they called their ride "Big Ellie". A search for this name lead us to believe that this Ferris wheel is a No. 5 Big Eli wheel made by Eli Bridge (I later emailed the park and confirmed that this is indeed the model of their Ferris wheel). Eli Bridge's website gave us some interesting information but not exactly what we were looking for. A bit more searching and we were able to find a pdf of the owner's manual for this ride that included a helpful diagram.

The diagram for this Ferris wheel shows that the height of the main axle to the ground is 22 feet, 3 and 3/16 inches. The height of the top seat to the ground is 39 feet, 11 and 9/16 inches. Subtracting these two heights gives us the radius of the Ferris wheel as 17 feet, 8 and 3/8 inches.

The ride manual also states that the proper operating speed for this wheel is 6 1/4 revolutions per minute.

It is only after we were able to answer some of the students' questions regarding the video of the Ferris wheel did we start to talk how we might mathematically modeling the height of a person riding the wheel over time and the periodic nature of this function. Students were much better able to make sense of this visual model once they had a good grasp of the context of the problem.